Journal of Geophysical Research: Space Physics
RESEARCH ARTICLE
MeV proton flux predictions near Saturn’s D ring
10.1002/2015JA021621
P. Kollmann1 , E. Roussos2 , A. Kotova2,3 , J. F. Cooper4 , D. G. Mitchell1 , N. Krupp2 , and C. Paranicas1
Key Points:
• Intensity between Saturn and D ring
is lower than in Saturn’s proton belts
• Intensity over the D ring might be
comparable to Saturn’s known belts
• Proton intensity is limited by
neutral density, not diffusion as in
known belts
Correspondence to:
P. Kollmann,
[email protected]
Citation:
Kollmann, P., E. Roussos, A. Kotova,
J. F. Cooper, D. G. Mitchell, N. Krupp,
and C. Paranicas (2015), MeV proton
flux predictions near Saturn’s D
ring, J. Geophys. Res. Space Physics,
120, 8586–8602,
doi:10.1002/2015JA021621.
Received 25 JUN 2015
Accepted 22 SEP 2015
Accepted article online 28 SEP 2015
Published online 24 OCT 2015
1 The Johns Hopkins University, Applied Physics Laboratory, Laurel, Maryland, USA, 2 Max Planck Institute for Solar System
Research, Göttingen, Germany, 3 Université Paul Sabatier Toulouse III, UPS-OMP, IRAP, Toulouse, France, 4 NASA Goddard
Space Flight Center, Greenbelt, Maryland, USA
Abstract
Radiation belts of MeV protons have been observed just outward of Saturn’s main rings. During
the final stages of the mission, the Cassini spacecraft will pass through the gap between the main rings and
the planet. Based on how the known radiation belts of Saturn are formed, it is expected that MeV protons
will be present in this gap and also bounce through the tenuous D ring right outside the gap. At least one
model has suggested that the intensity of MeV protons near the planet could be much larger than in the
known belts. We model this inner radiation belt using a technique developed earlier to understand Saturn’s
known radiation belts. We find that the inner belt is very different from the outer belts in the sense that its
intensity is limited by the densities of the D ring and Saturn’s upper atmosphere, not by radial diffusion and
satellite absorption. The atmospheric density is relatively well constrained by EUV occultations. Based on
that we predict an intensity in the gap region that is well below that of the known belts. It is more difficult to
do the same for the region magnetically connected to the D ring since its density is poorly constrained. We
find that the intensity in this region can be comparable to the known belts. Such intensities pose no hazard
to the mission since Cassini would only experience these fluxes on timescales of minutes but might affect
scientific measurements by decreasing the signal-to-contamination ratio of instruments.
1. Introduction
In this paper we study energetic protons very close to Saturn. Our focus is on spacecraft and instrumentpenetrating protons in the 10–100 MeV range but we extend our study down to keV energies. We consider
the gap region between Saturn’s dense atmosphere and the inner edge of the D ring, in the equatorial plane
between 1.02 and 1.06RS from Saturn’s center (Saturn radius RS = 60, 268 km). Also, the D ring extending out
to 1.24RS and magnetic field lines connected to this ring are considered.
During Saturn orbit insertion, the INCA camera, part of the MIMI instrument package [Krimigis et al., 2004] of
the Cassini spacecraft, was put in a unique location, from where it could detect energetic neutral atoms (ENAs)
with energies of tens of keV coming from the gap between the D ring and Saturn’s atmosphere [Krimigis et al.,
2005]. This measurement indicates the presence of trapped keV ions in that region. It is reasonable to assume
that this region is also populated by MeV ions similar to Saturn’s known ion radiation belts [Krimigis et al., 1983;
Armstrong et al., 2009; Roussos et al., 2011]. During closest approach, Cassini crossed field lines that map to an
equatorial distance of 1.42RS = 86 Mm in the equatorial plane, which lies in the C ring, near the Maxwell gap.
Energetic particle intensities measured by the MIMI/LEMMS instrument were low compared to the radiation
belts [Krimigis et al., 2005] (Figure 1). It is an active subject of research whether species and energies of the
particles are correctly determined or if the measurements result from cosmic ray or other contamination that
produces false detector counts. In comparison, the main telescope of the University of Chicago experiment
[Simpson et al., 1980] on board of Pioneer 11 counted MeV electrons and tens of MeV protons with rates that
were low but significant [Chenette et al., 1980; Cooper et al., 1985].
©2015. The Authors.
This is an open access article under the
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Attribution-NonCommercial-NoDerivs
License, which permits use and
distribution in any medium, provided
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KOLLMANN ET AL.
No spacecraft, including Cassini, has reached the radial distance of the D ring. Its inner part is almost completely transparent to light and so tenuous that its optical depth has not been constrained accurately [Hedman
et al., 2007]. It is therefore possible that MeV ions coexist with the ring, similar as how they survive in the E and
G rings.
The known ion belts are located between about 2 and 5RS , see black curve in Figure 1. They are collocated
with the orbits of several of the icy moons. Unlike at Jupiter, the magnetic and spin equators are coaligned,
meaning that the moons are very effective absorbers of energetic ions and isolate the belts from the rest of
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Figure 1. Compilation of measured and modeled radial intensity profiles. Black: Mission-averaged median
omnidirectional intensities of the MIMI/LEMMS P7 channel, measuring protons around 27 MeV, near the magnetic
equator. Error bars show the standard deviation. Green: Measurements during Cassini’s first orbit. We bin counts in
L shell and divide it by the integration time. Error bars are Poisson errors. Note detection of the Roche radiation belt
in the Roche division. Intensity over the rings is likely contaminated and an upper limit. (Pioneer 11 measured
10−8 ∕(cm2 sr s keV)) [Cooper et al., 1985]. Red: Best guess model of the radiation belt around the D ring for 27 MeV
protons as described through sections 2 and 3.
the magnetosphere [Roussos et al., 2008]. Therefore, the protons found in these belts cannot originate from
the magnetosphere or the solar wind but originate primarily from cosmic ray albedo neutron decay (CRAND)
[Singer, 1958; Hess et al., 1959; Krimigis and Armstrong, 1982; Cooper, 1983; Kollmann et al., 2013]. In this process, galactic cosmic rays (GCRs) [Blasi, 2013], typically > GeV protons, enter Saturn’s magnetosphere, collide
with material, the rings or planetary atmosphere, and create secondary neutrons at energies lower than the
primary particle. A fraction of these beta decay to protons, become trapped by Saturn’s magnetic field, and
populate the belts. Since the magnetic and electric fields fluctuate stochastically, the protons undergo a
stochastic motion that can be described well by radial diffusion [Roederer, 1970, pp. 112–132; Schulz and
Lanzerotti, 1974, pp. 81–92; Walt, 1994, pp. 92–146]. Diffusion leads proton trajectories to intersect moon
orbits, where they are absorbed. It is diffusion that causes a reduction in ion intensity even outside of the
radial range covered by the moon throughout its orbit.
The gap region between the D ring and Saturn and D ring itself are much closer to Saturn and the source
of neutrons than the known ion belts are. The magnetic field is stronger there, so that diffusive losses are
expected to be weaker. It has therefore been suggested that the inner radiation belt in the gap region has
a higher intensity of MeV protons than the known belts [Cooper, 2008; Cooper et al., 2011]. These estimates
were under the now questionable assumption of no absorption by atmospheric gas or ring material, which is
expected to be present.
The region that magnetically connects with the gap region and D ring is of special interest to the Cassini
mission. The proximal orbits scheduled for 2017 will lead the spacecraft over the rings and through the gap
(Figure 2). Particles bouncing along field lines that connect to the rings will reach Cassini even though it avoids
the rings themselves. Intense radiation on those field lines might affect Cassini’s instrument performance. For
example, the INMS instrument (measures neutral particles) [Waite et al., 2006] is contaminated by energetic
particles in the known radiation belts to an extent that studying this region becomes challenging [Elrod et al.,
2012]. Similar intensities in the inner belt could prevent in situ measurements of the exospheres of Saturn and
the D ring. If the intensities are extreme, they could even affect Cassini’s electronics and therefore interfere
with its operations.
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Figure 2. Cassini’s first orbit (black curve) in comparison with orbits planned for the end of the mission. We show a
typical proximal (red) and F ring (blue) orbit. L is the magnetic dipole shell, and z is the distance from the ring plane,
both in multiples of a Saturn radius. Vertical lines and shades indicate the locations of rings and gaps.
The purpose of this paper is to establish upper limits on the expected MeV proton radiation levels. We show
that the inclusion of atmospheric hydrogen gas brings down the intensity in the gap below the value present
in the known belts. The intensity connected with the inner D ring is more uncertain but likely not magnitudes
higher than the known belts.
This paper is structured as follows: We explain the model and our default assumptions in section 2. The relatively short section 3 presents the results. In section 4 we discuss how the results change depending on the
model parameters, some of which are poorly constrained so far. We find that our modeled spectra resemble
and help understand measurements in the magnetosphere. This is discussed in Appendix A.
2. Model
The model used in this paper is a modified version of the Kollmann et al. [2013] model, which we used to successfully reproduce Saturn’s known proton belts. We numerically solve either of the two following differential
equations for the phase space density, f , which is defined here as particles per volume in real and momentum
space (Note that this is different from a definition that uses velocity space.). Phase space density, f , is related
to the differential intensity, j, which is particles per time, energy interval, area, and solid angle, as f = j∕p2 with
the momentum p.
df𝜇,K (L)
̂
= S + Df
(1)
dt
dfL,𝛼 (E)
dt
̂
= S+̂
Ef + Cf
(2)
fx,y (z) means that, while f depends on x and y, these parameters are kept fixed and that the differential
equation describes the dependence of f on z. Specifically, 𝜇 and K are the first and second adiabatic invariants
[Roederer, 1970, pp. 19–23, pp. 46–52; Walt, 1994, pp. 36–50] (Table 1), and E is the particle kinetic energy.
The L shell gives the distance in multiples of RS of the equatorial crossing point of the magnetic field line occupied by the particle. The term 𝛼 is the equatorial pitch angle, which is the angle of the proton velocity vector
relative to the magnetic field direction at the magnetic equator. df ∕dt is the overall change of phase space
density per time. For simplicity, we assume a steady state: df ∕dt = 0. S is a source rate (gain of f per time).
̂ charge exchange. The ̂ symbol is used
̂ describes radial diffusion, ̂
Df
Ef particle loss by energy loss, and Cf
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Table 1. Explanation of Variables and Abbreviations Used Throughout This Paper
𝛼
equatorial pitch angle (angle of proton velocity relative to magnetic field at magnetic equator)
B
magnetic field
Bm
̂
Cf
magnetic field at mirror latitude 𝜆m
CRAND
cosmic ray albedo neutron decay
D
diffusion coefficient
charge exchange rate (loss of phase space density per time, equation 9)
dE∕dx
energy loss per distance in a given material (stopping power)
df ∕dt
̂
Df
overall change of phase space density per time (equations (1) and (2))
L shell diffusion rate (change of phase space density per time, equation (3))
E
energy
E0
reference energy for source (see section 2.1)
Ec
energy cutoff of galactic cosmic rays (minimum energy required by the Størmer limit to enter an L shell)
EE
̂
Ef
energy cutoff of source rate (EE =120 MeV)
ENA
energetic neutral atom
energy loss rate (change of phase space density per time, equation (4))
f
phase space density (protons per volume in real and momentum space)
GCR
galactic cosmic ray
H
vertical extent of D ring
I
integral proton intensity between 1 and 60 MeV, averaged over all directions (protons per time and area)
Imax
highest value of I measured at Saturn (Imax = 6 × 108 m−2 s−1 at L = 2.6)
j
K
differential intensity (protons per time, area, solid angle, and energy interval)
+𝜆 √
second adiabatic invariant K = ∫−𝜆 m Bm − B ds with ds along the magnetic field B
L
dimensionless dipole L shell (equal to radius in RS in equatorial plane, follows magnetic dipole to higher latitudes)
𝜆m
magnetic mirror latitude at which a bouncing particle turns around
m
exponent of source rate S
m
M
mass of one water molecule
Mm
106 m
𝜇
first adiabatic invariant 𝜇 = E(E + 2mc2 )∕(2mc2 B)sin2 𝛼
⟨n⟩
number density of neutral particles in D ring or exosphere, averaged over periodic proton motion (equation (6))
⟨n⟩r
number density of water molecules, averaged throughout a D ring with thickness H (equation (5))
n
exponent of diffusion coefficient D
n
D ring grain size distribution (ice grains of given radius per volume and radius interval)
p
proton momentum
PSD
phase space density (protons per volume in real and momentum space)
r
D ring grain size radius
RS
Saturn radius (60,268 km)
𝜌
mass density of water ice (mass per volume)
S
source rate (gain of phase space density per time)
𝜎
charge exchange cross section (area)
TB
bounce time
Tr
time a proton spends in D ring per bounce period
𝜏
optical depth of D ring (equation (7))
v
proton velocity
to mark operators that act on f , meaning that the result might depend on derivatives of f in L or E . Detailed
explanations of the terms are given below. Table 1 summarizes the major variables and abbreviations of
this paper.
Our solutions of equations (1) and (2) are computed by the commercial software Mathematica [Wolfram
Research, 2012]. We always check if the solution fulfills the differential equation and the boundary conditions.
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2.1. Source and Diffusion
The term S in equations (1) and (2) is the source rate. We describe it using the parameters that we determined
from the known radiation belts [Kollmann et al., 2013]. The source scales as S ∝ 1063 (3.5∕L)−m s2 ∕(kg3 m6 )
over L shell with an exponent m = 3. The dependence of the source on energy E goes as the measured ion
spectrum in 2 to 5RS region: for <8 MeV we use a power law S ∝ (E∕E0 )−3.9 with E0 = 1 keV. For higher energies,
we use a log-log scale parabola centered at 12.4 MeV with a rapid cutoff at EE = 120 MeV.
Equation (1) is the extreme case where the source is balanced only by radial diffusion that drives protons into
̂ describes diffusion of the phase space density f𝜇,K (L) in L shell.
other regions. Df
(
)
D 𝜕f
̂ = L2 𝜕
(3)
Df
𝜕L L2 𝜕L
D is the diffusion coefficient. We use D(L) = 0.3 × 10−9 (L∕3.5)n 1∕s with an exponent n = 10 [Kollmann et al.,
2013]. Equation (1) can be solved for fixed first and second adiabatic invariants 𝜇 and K . These quantities
depend on particle energy and pitch angle but, contrary to them, are constants during slow radial motions
[Walt, 1994, pp. 36–50].
2.2. Energy Loss and Charge Exchange
Equation (2) is the other extreme where radial diffusion is negligible and losses occur solely because of interactions of the protons with material. ̂
Ef represents the change in fL,𝛼 (E) of protons of a given energy E due to
energy loss in the atmosphere and in the D ring. Equation (2) can be solved for each L and equatorial pitch
angle 𝛼 separately. Atmosphere and ring interactions are each represented by a term of the following form
[Kollmann et al., 2013]
(
)
dE
v 𝜕
̂
p2 f
(4)
Ef = − 2
dx
p 𝜕E
where v is the proton velocity and p is the momentum. The energy loss per distance in a material dE∕dx is
available as tabulated values and does not have to be considered as derivative of a function. ̂
Ef can be either
positive or negative, i.e., causes either a decrease or an increase of the phase space density, depending on
the shape of the proton spectrum f and on dE∕dx (Results are shown in Figure 4.).
Equation (4) assumes that the protons undergo a continuous energy loss per traversed distance dE∕dx and
over their whole trajectory are subject to a constant density ⟨n⟩ (molecules per volume). We use values for
dE∕dx from Berger et al. [2005]. They scale approximately linearly with the density of the target particles even
if their impact parameter is larger than the nominal radius of the target molecule. A minor correction arises
because the ionization energy changes with target particle distance, but we neglect this here. The assumption
of continuous energy loss works well for a gas. As we show in Appendix B, it is also sufficient to describe
MeV protons interacting with mm to μm sized water ice grains. For lower energies or much larger grains, the
discrete energy loss and probability to hit a grain need to be taken into account [Kollmann et al., 2011].
For equatorially mirroring particles whose whole gyration is within the atmosphere, ⟨n⟩ is simply the density
at the equator. For bouncing particles, the density needs to be averaged over the trajectory. The distance to
Saturn’s center, h, of bouncing particles changes with latitude 𝜆 as h = Lcos2 𝜆. This change is much stronger
than Saturn’s oblateness (polar radius is 0.90RS ) [Seidelmann et al., 2007] or the atmospheric asymmetries
around the exobase (< 0.01RS ) [Koskinen et al., 2013]. More field-aligned particles will therefore experience
stronger losses, yielding lower intensities. This means that the pitch angle distribution of particles whose
motion is completely within the atmosphere will peak for equatorially mirroring particles. Since we are aiming
for upper limit intensities, we will use the equatorial density for all pitch angles.
For the D ring, we assume that the molecules of the ice grains are homogenously distributed over the ring.
The molecular density ⟨n⟩r of the ring is related to the grain size distribution n (grains of radius r per volume
and radius interval) as
rmax
𝜌
4𝜋 3
⟨n⟩r =
nr dr
(5)
M ∫rmin 3
with the mass density 𝜌 of water ice and the mass M of a water molecule. The different kinds of densities used
throughout this section are summarized in Table 1. We assume that n is spatially constant throughout the
D ring with vertical extend H across the equatorial plane. The density averaged over the proton trajectory is
then
T
⟨n⟩ = ⟨n⟩r r
(6)
TB
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with the time Tr spent in the ring and the bounce time TB . Note that this is different from the ratio of latitudes.
This expression can be related to the optical depth 𝜏 , which can be calculated as
𝜏=
Forcing equation (7) into (6) yields
rmax
∫rmin
nH𝜋r2 dr
(7)
r
⟨n⟩ =
max 3
̃
4𝜌𝜏Tr ∫rmin r ndr
rmax 2
3HMTB ∫
̃
r ndr
r
(8)
min
̃ . The advantage of equation (8) over
The grain size distribution can be expressed as a product: n = n0 n(r)
(6) is that it only depends on the shape ñ of the size distribution, not its absolute value. The density ⟨n⟩ has
only a weak explicit dependence on the ring thickness H since Tr ∝ H is approximately true for a thin ring.
(The ring thickness is implicitly included via the optical depth 𝜏 .) We assume here H = 100 m, about an order
of magnitude more than the A ring [Colwell et al., 2009a].
The most field-aligned equatorial pitch angle at the outer edge of the D ring is 40∘ . Particles with smaller pitch
angles have their mirror point with the dense atmosphere and are lost. We calculate the density ⟨n⟩ for this
angle. More equatorial particles will experience higher densities since they will spend more time Tr within the
ring, resulting in lower intensities.
An exception is particles that mirror very close to the magnetic equator. Since the magnetic equator is offset from the ring plane (by 0.036RS ) [Burton et al., 2010], these particles will not suffer strong losses. In time,
they can lose energy from interaction with the ring exosphere. At low energies, pitch angle scattering in the
exosphere will become important and eventually drive them into the ring. If this is slow compared to the
source rate, particles will accumulate to high intensities. Since these protons will by definition not bounce up
to Cassini’s latitude, they cannot be measured in situ or cause problems to Cassini or its instruments. In principle, they might be detectable remotely as ENAs. MIMI/INCA will measure them in case the intensity at tens
of keV is high enough.
On top of energy loss, interaction with gas and ice scatters the protons (causing pitch angle diffusion) and
broadens their energy distribution (straggling, causing energy diffusion). We do not consider this here since
it turns out that this is negligible for MeV protons [Kollmann et al., 2013].
̂ accounts for charge exchange. This process converts protons into energetic neutral atoms (ENAs) that
Cf
escape the Saturn system.
̂ = −𝜎⟨n⟩vf
Cf
(9)
where 𝜎 is the charge exchange cross section. We use values for protons charge exchanging with H from
Lindsay and Stebbings [2005] and of H2 from Barnett et al. [1990]. They provide values extending to about
220 keV, and we extrapolate beyond that energy. The error from this extrapolation does not play a role since
at these energies charge exchange becomes negligible compared to energy loss.
Charge exchange is only important for energies below several 100 keV and is only included for completeness.
It is possible that the charge exchange losses are compensated or even countered by another process: ENAs
can be stripped of an electron again when colliding with a neutral particle [Barnett et al., 1990]. The ENAs do
not have to be produced locally but can also originate from other locations in the inner belt or the magnetosphere. It was suggested that the main source of keV ions in the gap region is from stripping [Krimigis et al.,
2005], similar to Earth’s low-altitude proton belt [Moritz, 1972; Gusev et al., 2003].
Protons passing ice grains can become neutralized [Kreussler and Sizmann, 1982]. This additional sink is not
considered here.
2.3. Atmospheric and D Ring Model
The gas in the gap region mainly originates from Saturn’s atmosphere but might have contributions from
the rings. The region we are considering include parts of the thermosphere and exosphere, separated by the
exobase around 1.05RS [Koskinen et al., 2013]. We use values that approximate the Cassini project’s engineering atmosphere model initially developed in 2010 (D. Strobel, private communication). It was subsequently
validated by solar EUV occultations described in Koskinen et al. [2013] and stellar EUV occultations in [Koskinen
et al., 2015], all observed with the UVIS instrument on Cassini [Esposito et al., 2004]. The dominating species
close to Saturn is H2 . We assume an equatorial density of 4 × 1016 m−3 at an altitude of 1.02RS . Inward of this
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altitude the density rises steeply, outward
it decays exponentially to 2 × 1014 m−3
at 1.05RS . Outward of this distance, H
is dominating and also decays exponentially with about the double scale length.
The optical depth of the outer D ring,
between 73 and 74.5 Mm (1.21–1.24 RS )
[Horányi et al., 2009], coincident with
the D73 ringlet, is of the order of 10−3
[Hedman et al., 2007]. We calculate the
average density ⟨n⟩ using equation (8)
and assume a grain radius distribution
that scales as n ∝ r−3 between 0.1 μm and
1 mm and is zero otherwise. We will discuss different densities in section 4 since
the size distribution is very uncertain.
Figure 3. Model results for different processes included. (a) Radial
intensity profile of 27 MeV protons. (b) Energy spectra at L = 1.03.
Green: Assuming only radial diffusion and no material in and around
the D ring. Blue and Orange: Assuming exospheric gas densities that
approximate the Cassini project’s engineering atmosphere. Blue
considers only energy loss, orange also accounts for charge exchange.
Diffusion is negligible here when these effects are present. Spectra in
the D ring are not shown but resemble the blue curve. (There is no
difference in Figure 3a between the blue and orange profiles since they
are at an energy not affected by charge exchange.) Red: Additionally
including energy loss in the D ring ice grains. (No difference in
Figure 3b between red and orange spectra since they are for a location
of no significant D ring density.)
The optical depth of the inner part
of the ring, between 67 and 73 Mm
(1.11–1.21 RS ), is so low that it has not
been well constrained to date [Hedman
et al., 2007]. We assume here that the
density ⟨n⟩ in this region is 10 times lower
than in the outer D ring. We neglect the
embedded brighter D68 and D72 ringlets
here. Inside of 67 Mm, the normalized I∕F
reflectance shows an exponential decay
with scale length 1000 km (M. M. Hedman, private communication, 2013). We
assume that the I∕F profile is proportional
to the density. Although I∕F flattens
out inside of about 64 Mm (1.06RS ), we
assume that it keeps falling in order to
get upper limit intensities.
We discuss further details of the atmosphere and D ring densities together with
their uncertainties in sections 4.1 and 4.2.
2.4. Boundary Conditions
Equation (1) requires two boundary conditions. We assume f = 0 at L = 1.02,
where the atmospheric density starts to
rise steeply, and at L = 1.24, about the
outer edge of the D ring. Equation (2) requires one boundary condition. We use f = 0 at EE = 120 MeV, which
is the energy where we force the source to zero (section 2.1). The value of EE does not affect the results at
energies well below it.
3. Results
3.1. Best Guess
Modeled radial intensity profiles and energy spectra using the most likely parameters can be found in Figure 3.
How the model changes depending on these parameters will be discussed in section 4. Here we discuss how
different physical processes affect the model results.
The green curves in Figure 3 represent the theoretical case with no material present. Figure 3a shows a radial
profile at a fixed energy (near the CRAND peak), and Figure 3b an energy spectrum at a fixed location (between
the dense atmosphere and the D ring). The radial profile has its maximum about in the center of the region
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considered and is forced to zero by
strong losses at the edges. (The maximum
appears as a plateau since Figure 3 spans
many orders of magnitude, but the contrast between the peak and the regions
just outside where the steep losses occur
is about a factor of 10.) Radial diffusion
drives the protons to these edges and
therefore limits the overall intensity. Since
diffusion is treated as energy independent, the energy spectrum has the same
shape as the source function.
The orange and blue curves assume a gas
density that approximates the engineerFigure 4. Absolute value of the change of intensity per time
ing atmosphere model. It can be seen in
2
(p |df ∕dt|) for the best model. Thick violet: Source rate ∝ S. Dashed
Figure 3a that the intensity increases radĩ from charge exchange. Dashed blue: Intensity
orange: Loss rate ∝ Cf
̃ ) in the atmosphere. The
ally outward as the gas density decreases.
change resulting from energy loss (∝ Ef
intensity increases between 100 and a few hundred keV and decreases
The blue spectrum in Figure 3b includes
otherwise. Sign changes show as sharp dropouts. Dashed red: Energy
only energy loss in the atmospheric gas.
̃ . Black: Proportional to
loss in the D ring. Green: Diffusive loss rate ∝ Df
Energy loss washes out the CRAND peak
the intensity spectrum for comparison.
around 10 MeV and generally makes the
spectrum flatter compared to the loss-free spectrum (green curve). The orange curve shows the effect of
adding change exchange. It causes depletion below a few hundred keV, where the charge exchange cross
section becomes larger. This depletion resembles the spectra that are measured inside of about L = 8 in
the magnetosphere. Although this seems to suggest that the magnetospheric spectra are shaped by charge
exchange, we have not found sufficient evidence to support this theory, as discussed in Appendix A.
Lastly, the red curve in Figure 3 shows the intensity computed when adding the D ring and the energy loss
associated with it are included. The ring limits the intensity at large distances from Saturn (Figure 3a). Especially the dense outer part of the ring strongly reduces the intensity. The model shows an intensity maximum
around 1.08RS , where both the atmosphere and the D ring have a relatively low density. Unfortunately, the
model is least reliable at this distance since it depends about equally on atmospheric and ring effects, while
at other distances only one is dominant.
Figure 4 shows the relative importance of the different physical processes as a function of energy. The radiation
belt around the D ring turns out to have very different characteristics from those of the known belts. The
main loss process in the known belts is radial diffusion, driving the protons to the moon orbits . Losses in the
Neutral Torus around Enceladus’ orbit only play a minor role [Kollmann et al., 2013]. Diffusion turns out to be
negligible in the inner belt. Losses are dominated by the interaction with the material. The figure shows the
gap region, where this material is largely atmosphere. Energy loss dominates above a few hundred keV and
charge exchange below. Since we assume a steady state, these losses balance the source rate exactly. In many
cases, energy loss is causing a net particle loss at a given energy (̂
Ef < 0). This is different between 100 and
400 keV, where the spectrum is rising steeply with energy, energy loss is causing a net gain of those protons
(̂
Ef > 0).
3.2. Comparison With Known Radiation Belts
In order to simplify the discussion on how the model changes with its parameters we reduce the modeled
proton distribution over distance and energy to a single parameter, namely the integral intensity I = ∫ j dE dΩ,
describing the number of protons in an energy range per time and area. We use 10 MeV as the lower energy
of the integral since this is a typical value where proton penetration starts to become important. We truncate
the integral at 60 MeV because we base our source strength on measurements by the LEMMS instrument
[Kollmann et al., 2013], which provides constraints only up to this energy.
Cassini will pass close enough to the planet that the loss cone size (usually very small) becomes relevant,
i.e., that the intensity of field-aligned particles becomes almost zero because they bounce into the dense
atmosphere and are lost. On the other hand, Cassini remains at high latitude for most of the proximal orbits,
so that particles mirroring at lower latitudes do not reach Cassini’s location. It turns out that the maximum
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solid angle with significant intensities during the
proximal orbits is ∫ dΩ = 2𝜋 . For the calculation
of I, we assume that j is constant over this range
and zero otherwise.
The default values of our model yield I = 70 ×
105 m−2 s−1 at the peak of the radiation belt
(L = 1.08). It is lower in the middle of the D ring
(10 × 105 m−2 s−1 at L = 1.03) and the gap region
(0.1 × 105 m−2 s−1 at L = 1.03 and 4 × 105 m−2 s−1 at
L = 1.05). How this changes with the parameters
is discussed in section 4.
Figure 5. Energy spectra of the model compared with
measurements. Green, orange, and red: Modeled spectra
assuming the best guess model at distances given in the
legend. L = 1.05 is an area of relatively high and reliable
intensity. (Since the gas density is low and the D ring does not
have a strong effect yet.) Blue and cyan: Mission-averaged
measurements with the MIMI instrument at locations given in
the legend. (We use omnidirectional measurements near the
magnetic equatorial plane and average them linearly after
applying a median filter. The blue curve shows MIMI/LEMMS
alone, the cyan curve a combination with MIMI/CHEMS.)
L = 2.67 is the region of the most intense radiation observed
so far at Saturn, between the orbits of Janus and Mimas. L = 7
is an example magnetospheric spectrum. Black: 1 count level
of LEMMS when integrating over 240 s.
The most intense proton radiation at Saturn is
found in the radiation belt centered at L = 2.7,
between the orbits of Janus and Mimas and
close to the G ring, see Figure 1. This was passed
about 14 times by Cassini to date. We interpolate
the mission-averaged median measurements
in energy and integrate over the energy range
given above. This yields Imax = 6 × 108 m−2 s−1 ,
a value 90 times larger than the default model.
When Cassini flew by Earth, it encountered the
highest intensities of >10 MeV energies at a radial
distance of 1.4RE . We linearly fit the energy and
integrate it over the given range. This yields a
slightly higher value of 10 × 108 m−2 s−1 . It should
be noted that this value is not typical for Earth.
We only determine it since Cassini passed Earth
without issue indicating that this intensity is safe,
at least for a single pass. Cassini was designed
to withstand even harsher environments with
500 × 108 m−2 s−1 (Cassini Orbiter Functional
Requirements Book, Environmental Design
Requirements, November 1996).
While it is possible that the intensity around the D ring is comparable to the known belts, it is also possible
that it is too low to be detected at all. A lower limit for detection is the 1 count level, i.e., assuming that an
instrument channel receives 1 count while passing this region. A statistically significant signal needs to be
well above this. Another difficulty will be determining if the measurement is foreground or contamination
from Cassini’s radioisotope thermoelectric generators or galactic cosmic rays. We calculate the 1 count level
here for a period of 4 min and show it in Figure 5. Four minutes is slightly smaller than the average time that
Cassini will be connected to the gap region per orbit. The time connected with the D ring is about 5 times
larger, causing an equally smaller 1 count level. It can be seen that our best guess intensity for MeV energies in
the D ring and the outer edge of the gap region (1.05 ≲ L < 1.24) is an order of magnitude above the 1 count
level, meaning that LEMMS might be able to detect these protons. Based on the uncertainties of the model,
we consider it as possible that LEMMS will detect MeV protons also at smaller distances.
4. Parameter Study
Our model predicts for the gap region an intensity well below Imax but still above the 1 count level. The intensity in this region depends on the atmospheric density and the source strength, which are relatively well
known. Realistic changes in the parameters should not change the result by much more than an order of
magnitude. We discuss the details below.
The region magnetically connected with the D ring has much higher intensities that can be comparable to
Imax . The intensity in this region is known with less certainty than the gap region since the D ring density is
poorly constrained. It is easy to justify densities that change the intensity by more than 3 orders of magnitude.
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4.1. D Ring Density
The largest uncertainty of our model is
the density of the D ring. There is only an
upper limit on the optical depth of most
of the ring (<10−3 inside 1.21RS ) [Hedman
et al., 2007], and the grain size distribution necessary to relate it to its mass density is not well known either. Grain size
distributions are commonly power laws
roughly following n ∝ r−3 [Charnoz et al.,
2009] over a range of 0.1 m < r < 10 m
for the main rings [Cuzzi et al., 2009] and
1 μm < r <100 μm for the tenuous rings
(M. M. Hedman, private communication,
2014). The power law will break down at
small sizes since small grains can stick to
Figure 6. Omnidirectional proton intensity integrated between 10 and
larger grains or leave the Saturn system
60 MeV. Black curve: Model intensities for different D ring number
due to their charge and the corotational
densities ⟨n⟩ at the radial maximum of the radiation belt. The number
19
3
2
electric field. The power law also breaks
density 10 ∕m converts to 3μg/cm column density, which is
down at large sizes depending on the
insignificant compared to the tens to hundreds of g/cm2 of the A and
B rings [Colwell et al., 2009b; Robbins et al., 2010]. Realistic values for
grain origin. The locations of these breakthe D ring should not be too far off the two vertical lines (section 4.1).
downs are not well known since in optiThe red vertical line marks the density resulting from an optical depth
cal measurements, the wavelength range
of 10−4 and grains of 1 μm size. The green vertical line assumes the
constrains the range of grain sizes those
same optical depth but a power law grain size distribution. The line
wavelengths are most sensitive to. The
labeled Saturn maximum is the highest intensity so far measured at
Saturn (section 3.2). Earth maximum is the highest value Cassini
mass density is insensitive to the lower
experienced during the Earth flyby. Cassini maximum is the intensity
0.1
) but sensitive to the
cutoff (⟨n⟩ ∝ rmin
Cassini is designed to withstand. Zero density is the extreme case of no
0.9
upper
one
(
⟨n⟩
∝
rmax
). Lowering the
D ring and atmosphere. The intensity is in this case limited by radial
upper
cutoff
by
an
order
of magnitude
diffusion. The intensity for infinite density is from transient protons
will reduce the ring density by almost the
before encountering the ring (section 4.5).
same amount. The result is not strongly
dependent on the details of the size distribution. Assuming that the size distribution has a sudden change in
slope only changes the density by about a factor of 0.5.
Hedman et al. [2007] inferred that the size distribution is narrow. In the outer D ring, it is dominated by
10−100 μm grains while the inner D ring shows 1 μm grains. Assuming that there are only 1 μm grains yields a
density that is 100 times lower than what is assumed as our best guess. Using a more realistic Hansen-Hovenir
distribution [Hansen and Hovenier, 1974; Hedman et al., 2007] instead of a single grain size only changes the
prediction by about a factor of 2.
Figure 6 shows the effect of different ring densities on the integral intensity I. It can be seen that our default
case (r−3 size distribution up to 1 mm) yields intensities 100 times below Imax while the case of a narrow
distribution of 1 μm grains yields intensities that are comparable to Imax .
4.2. Atmospheric Density
The atmospheric model was designed to determine the altitude at which the Cassini spacecraft starts to tumble due to atmospheric drag during its last orbit. We use the model in the range where it is supposed to be
valid: Tumbling should occur around 1.02RS . Koskinen et al. [2013] used EUV occultation data to determine the
exobase at 1.05RS and the H2 density up to about 1.06RS .
The engineering model uses an upper limit of the H density. Despite being a small fraction of the number
density near Saturn [Koskinen et al., 2013], H becomes highly dominant outside of 1.05RS due to its slower
falloff with radial distance. If we assume instead that H is always negligible, I increases by a factor of 2.
The overall error of the engineering model near the equator is estimated to be smaller than a factor of 2.
This is excellent compared to the uncertainties of the other parameters. However, the atmosphere expands
and contracts over time [Koskinen et al., 2015] and the density above 1.06RS has not been directly measured.
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We therefore consider a change in the
exponential decay length of H2 outside
of 1.02RS . How this affects I is shown in
Figure 7. A fast decay causes I to increase
near Saturn, but not to a level that would
be comparable to the D ring. A slow decay
on the other hand will bring down the
intensity in the D ring. The atmosphere is
therefore not critical for determining the
maximum possible intensity I.
4.3. Source
Kollmann et al. [2013] determined the
radial power law exponents n and m of
the diffusion coefficient and source rate
(S ∝ L−m ) in the explored radiation belts.
A good fit to the observations required
Figure 7. Omnidirectional modeled proton intensities for different
n + m ≈ 14 and m ≈ 3 but was not senscale heights of the thermosphere. Red: Intensity in the gap region, at
R = 1.03. Green: Maximum intensity in the D ring.
sitive to the exact values. The results of
the current model depend strongly on the
scaling of the source and its exponent m. Figure 8 illustrates that an exponent of m ≈ 7 would cause (with the
default D ring density and other parameters) an overall I that is comparable to Imax . Such a steep and steady
rise is not expected from theory. Theory suggests smaller exponents than we are using. Assuming a low exponent of m = 1 or that the source remains constant inside of the distances covered by the Kollmann et al. [2013]
model yields intensities 1–2 magnitudes below our default case.
We first consider the conceptual cases of isotropic emissions from a sphere or a ring without the presence
of a dipole field. The neutron differential intensity j (particles per solid angle, area, time, and energy interval)
is constant along the neutron trajectories, which are almost straight lines after leaving the material in which
they were created from GCRs. The intensity does not change significantly here because the neutron lifetime
is long compared to the time spent near Saturn and the environment around Saturn’s atmosphere is tenuous.
The omnidirectional intensity (particles
per area, time, and energy interval) calculated as J = ∫ jdΩ where the integration
includes the solid angle covered by the
source at any given location. Assuming
isotropic, homogenous emission from a
sphere of radius R yields immediately
J = 2𝜋j(1 − Cos{Asin[R∕r]}) as a function
of distance r to the center. It is J≈𝜋jR2 ∕r2
for r → ∞ but steepens for r → R, which
causes an underestimation of J if a 1∕r2
extrapolation is used. The solid angle of a
disk as a function of arbitrary location was
calculated by Paxton [1959]. J of a ring can
be calculated by subtracting the values of
a disk with small radius Ri from a disk with
large radius Ro . This yields J ∝ 1∕r2 for
Figure 8. Omnidirectional modeled proton intensities for different
r ≫ Ro and J ∝ r for r ≪ Ri with the
radial exponents m of the source. Red: Intensity in the gap region, at
distance r from the ring center. This is
R = 1.03RS . Green: Maximum intensity in the D ring. Red vertical line:
Exponent of the known radiation belts (> 2.3RS ). The value of m
generally true for all latitudes. Observachanges between neighboring plus and cross symbols by 1. Blue
tion near the ring plane shows an addivertical line: Source rate modeled outside of the rings, at 2.67RS
tional, relatively strong gradient when
[Kollmann et al., 2013]. Green vertical line: The theoretical exponent m
approaching the ring edges and flat proranges between 1 and 2 (section 4). Black vertical line: Modeled
exponent [Kollmann et al., 2013].
file over the ring.
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A proper calculation needs to take into account the magnetic field and provide J as a function of L, not r.
All protons created from neutrons on any point of a magnetic field line contribute to the source as long as
they have pitch angles that do not correspond to mirror locations deep in the atmosphere. Cooper [1983] and
Dragt et al. [1966] calculated omnidirectional proton intensity (particles per area, time, and energy interval)
for different cases of interest, and we fit their results with power laws in L.
Isotropic neutron emission from a sphere scales with an exponent between 1 and 2 at least in the range 1.2 <
L < 4. Anisotropic or inhomogeneous emission mostly from the poles yields an exponent in the same range. It
is possible that most neutrons are emitted from Saturn’s atmosphere, which can be considered as a spherical
source.
There can be important neutron contributions from the rings. Isotropic emission from a flat ring (radial extent
from 1.5 to 2.3RS ) is not described well with a power law over a large L range. Exponents just outside the ring
range between 3 and 5. The intensity profile is flatter farther out and above the ring so that extrapolation using
the given exponents yields an upper limit. Anisotropic emission modeled from the CRAND process yields,
outside the rings (2.3 < L < 4), exponents of 4 for > 15 MeV protons and 6 for > 600 MeV [Cooper, 1983].
The source rate for transient protons over the rings (L < 2.3) can be estimated to actually decrease with an
exponent of −2: Only galactic cosmic rays with energies above the Størmer cutoff (Ec ∝ L−2 for > GeV protons)
[Sauer, 1980; Smart and Shea, 2005] reach the rings. The proton production from cosmic ray protons on oxygen
nuclei scales roughly as S ∝ Ec−1 [Light et al., 1973] and gives rise to a steady state intensity j ∝ S ∝ L2 ,
consistent with Pioneer observations [Chenette et al., 1980].
4.4. Diffusion
Since diffusion is negligible inside the D ring compared with the strong losses in the neutral material, the
diffusion exponent n is not expected to play a major role. Only if the exponent becomes as small as n ≈ 1, the
̂ for the solution of equation (2)) becomes comparable to the other rates. Diffusion will act as
diffusion rate (Df
an additional loss process. It will change the radial profile of the belt and generally decrease its intensity.
4.5. Limiting Cases
In case of unrealistically low densities of the D ring and atmosphere, the intensity will be limited by radial
diffusion, as illustrated in Figure 3.
More realistic is the other extreme of a D ring that is sufficiently dense that any proton created is stopped in
the ring during its first bounce along the magnetic field lines. This is what happens in the A to C rings. The
way we implement source and loss in our model will yield an intensity that approaches zero (Figure 6). This
is due to the implicit assumption that energy loss is continuous. In reality, the protons will move freely until
the point where they encounter the ring. The intensity of this transient proton population is independent of
the ring density, solely determined by the source rate S and the bounce time TB and can be estimated as STB .
The result is added to Figure 6. It is small but nonzero.
5. Summary and Conclusions
1. The major loss processes for protons with energies above hundreds of keV in the inner radiation belt are
energy loss in the material of the D ring and the planetary atmosphere. Radial diffusion is minor. This is
opposite to the known radiation belts.
2. We compare the modeled 10–60 MeV integral intensity with the maximum intensity Imax experienced so
far at Saturn. The modeled intensity at the outer edge of the gap region (1.05RS ) is 3 orders of magnitude
below Imax and at MeV energies 1 magnitude above the detection threshold of the LEMMS instrument. The
uncertainties in this region should not change the result by much more than an order of magnitude.
3. The proton intensity on magnetic field lines connected to the D ring can reach Imax for not unreasonable
assumptions about the D ring. This intensity should still pose no harm to the spacecraft, especially given
the shortness of the exposure times. However, the inner belt of MeV protons could affect scientific measurements, for instance, by contributing background counts in detectors. Due to large uncertainties, it is possible
that the intensity is orders of magnitude below Imax .
4. Energetic particles might accumulate over the rings at the magnetic equator. These do not bounce to higher
latitudes and can therefore only be detected indirectly, e.g., by the remote detection of energetic neutral
atoms.
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Figure A1. (left) Spectra of protons and (right) water group ions in Saturn’s magnetosphere. Water group ions (Hn O+
with 0 ≤ n ≤ 3) are dominated at these energies by O+ so we assume them to be oxygen ions. Measurements by the
MIMI/CHEMS instrument (cross symbols) and by MIMI/LEMMS (asterisk symbols) of all pitch angles taken within ±10∘
latitude are linearly averaged throughout the Cassini mission after applying a median filter. The smooth light blue curve
in Figure A1 (left) is a model result, and the green curve is the assumed source rate (intensity provided per time). The
red curves in both panels show the charge exchange rate (cross section times ion velocity) with water gas. The feature
around 100 keV resembles the modeled spectra of the inner belt (Figure 5). It is weaker for oxygen ions when comparing
to proton spectra taken at the same distance (see labels). Arrows indicate how the spectral feature is caused by the
source and loss processes. The background color marks the three energy ranges that are shown in Figure A2 with similar
color coding. Note that instantaneous spectra usually only show a peak around 100 keV. Intensities at lower energies can
be in more than half of the cases zero within about L = 6. Averaging these zero values with the large, nonzero intensities
that occur throughout the mission yields the low-energy power laws shown here. We did not find a consistent change
of the spectra with pitch angle but such a dependence might be hidden by the strong variability over the mission.
Appendix A: Charge Exchange in the Magnetosphere
Introducing charge exchange into the model causes an intensity decrease around 100 keV followed by a
spectral peak (Figure 3, orange curve). This spectral shape is reminiscent of what is measured in the magnetosphere (Figure A1, left), suggesting that this feature is also caused by charge exchange. The feature in the
spectrum is related to a feature of the radial intensity profiles shown in Figure A2. The radial profiles show
a peak around L = 9 for < 60 keV. Above these energies, approximately where the charge exchange cross
sections for protons on the gas of the Enceladus neutral torus falls off, the peak is shifting inward to about
L = 7. This shift in the radial peak is causing the peak in the spectrum. Understanding the spectra is therefore
helpful in understanding the radial distribution.
We solve our model equation (2) for the magnetosphere. The source in this case is not CRAND but radial
inward transport from larger distances via radial diffusion (section 2), interchange, and/or dipolarization
events [Arridge et al., 2011]. We assume a source spectrum with a shape that resembles the measured spectrum at larger distances. The gas in this region is from the distant portions of the Enceladus neutral torus,
which consists predominantly of H2 O and O, depending on distance [Cassidy and Johnson, 2010]. We assume
charge exchange cross sections for H2 O [Toburen et al., 1968; Dagnac et al., 1970; Greenwood et al., 2000; Gobet
et al., 2001], but using O yields similar results. Since the overall source strength is a free parameter here, the
result is independent on the assumed neutral density and will only reproduce the spectral shape and not the
absolute intensity. The model result shown in Figure A1 roughly agrees with the measured spectrum. The gradient change below 100 keV results from the drop of the charge exchange rate 𝜎v around that energy. The
second gradient change at slightly higher energies results from the shape we gave the source spectrum S.
While we can reproduce the proton spectra using plausible assumptions, this does not work for the oxygen
ion spectra shown in Figure A1, right. The charge exchange rate of oxygen ions is approximately constant over
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Figure A2. Proton intensity profiles as a function of L shell distance to Saturn. Different colors are different energy
ranges that use the same color coding as the background shading of the spectra in Figure A1 and are explained further
in the legend. Intensities measured throughout the Cassini mission within ±10∘ latitude are linearly averaged after
applying a median filter. Data were taken by the CHEMS and LEMMS instruments. Smooth curves show polynomial fits
to the data that were used to determine the radial intensity peak (black solid diamonds) for each energy. While at large
distances, low energies (red) have the highest intensities, the shift in the radial peak is causing that large energies (blue)
inward of about L = 7 have higher intensities than smaller energies (green). This is causing the 100 keV peak in the
spectra of this region (Figure A1). Filtering the data for different azimuthal locations shifts the peaks in L shell [Kollmann
et al., 2011; Thomsen et al., 2012] but roughly the same way for all energies.
the energy range considered here. Reproducing the spectra with a model as above requires that the source
alone would need to be responsible for the steady state spectrum. Even if this were a good model, it would
mean that the spectral shape is not determined by charge exchange.
There is evidence that energetic protons and oxygen ions are indeed governed by different physics: Dialynas
et al. [2009] were fitting the spectral peak that is roughly at 100 keV and found that its precise location in
energy changes adiabatically with distance to Saturn in case of protons but not of oxygen. However, proton
and oxygen spectra are very similar. It is therefore unlikely that charge exchange is the dominant process
shaping the proton spectra while a different process is responsible for the oxygen spectra but still results in
the same spectral shape. If we doubt the importance of charge exchange for oxygen ions, we therefore need
to do the same for protons even though their spectra can be reproduced under this assumption.
It is possible that our inability to establish the importance of charge exchange is because our model is too
simplistic. Further analysis should be a topic of further investigation. For now, we would like to point out
that even though charge exchange is ongoing (because it is source for the ENAs imaged by MIMI/INCA)
[Krimigis et al., 2005, 2007], it does not mean with certainty that it is also the dominant process in shaping the
ion distribution.
Appendix B: Discrete Versus Continuous Energy Loss
The change of phase space density at a given energy for particles suffering a continuous energy loss without
scattering of their velocity is described in equation (4). An equivalent expression can be found without derivation in Schulz and Lanzerotti [1974, equation 2.16, p. 56], and a derivation is given in Kollmann et al. [2013].
We consider the other extreme here, namely discrete energy loss that only happens when a projectile hits
a target. It is assumed that all targets are spherical grains with diameter X̃ . The average distance traversed
̃ since not all grains are hit centrally. The change of proton phase space
through each sphere is X = 2X∕3
density f at a given energy Ei per time is determined by (1) the source rate S at this energy, (2) the loss
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Figure B1. Intensity of protons that are injected into a material with a given spectrum (green curve) and suffer energy
loss. Black curve: Resulting proton spectrum if the material is water gas and the protons suffer continuous energy loss.
Red, orange, and blue symbols: The water is distributed over ice grains with sizes given in the legend. The distance
between large symbols of the same color is due to the discrete energy loss in the ice grains. Energies in between the
symbols cannot be reached if the starting energy is EE = 120 MeV. The small dots filling the gaps use slightly different
values of EE .
Acknowledgments
Cassini/MIMI data are available online
through NASA’s planetary data system
(PDS). The JHU/APL authors were
partially supported by the NASA
Office of Space Science under task
order 003 of contract NAS5-97271
between NASA/GSFC and JHU. The
MPS authors were partially supported
by the German Space Agency (DLR)
under contract 50 OH 1502, by the
Max Planck Society, and by the
International Max Planck Research
School for Solar System Science
(IMPRS) at the University of Göttingen,
Germany and at the Max Planck
Institute for Solar System Research
(MPS). The authors like to thank D.
Strobel (JHU) for developing the
engineering atmosphere model, M.M.
Hedman (Cornell University) for his
analysis on the D ring, A. Lagg (MPS)
for analysis software support, and
J. Vandegriff (JHU/APL) and M. Kusterer
(JHU/APL) for data reduction.
Michael Liemohn thanks Nick Sergis
and one another reviewer for their
assistance in evaluating this paper.
KOLLMANN ET AL.
rate from protons encountering a target grain and losing any nonzero energy in it, and (3) the source from
particles with a higher energy Ei−1 that encounters a target and loses Δ = Ei−1 − Ei . Expressing this as an
equation gives
df (Ei )
= S(Ei ) − ⟨n⟩𝜎vi f (Ei ) + ⟨n⟩𝜎vi+1 f (Ei−1 )
(B1)
dt
where ⟨n⟩ is the grain number density, 𝜎 is the geometric grain cross section, and v is the proton velocity (much
larger than the grain velocity). This means that f (Ei ) can be iteratively calculated after choosing a boundary
condition at i = 0. We pick f = 0 at E = 120 MeV in analogy with equation (2). The energy loss Δ is calculated
numerically based on the average thickness and the energy loss in the grain dE∕dx :
X
Δ=
∫0
dE(E)
dx
dx
(B2)
We show example solutions for different grain sizes in Figure B1 and compare them to equivalent solutions
of equation (2) for continuous energy loss. It can be seen that at high energies, where the energy loss per
encounter is small, all solutions fall on top of each other. This is the energy range that is of most interest in
this paper. The solutions differ from each other at the lower energies. Larger grains cause larger differences.
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